In wireless communications employing coherent detection, imperfect knowledge of the fading channel state imposes limits on the achievable performance as measured by, e.g., the mutual information, the bit-error rate (BER), or the minimum mean square error (MMSE). Typically, a fraction of system resources—bandwidth and energy—is devoted to channel estimation techniques (known as training) which improve knowledge of the channel state. Such schemes give rise to a tradeoff between the allocation of limited resources to training on one hand and data on the other, and it is natural to seek the optimal allocation of resources between these conflicting requirements. Such an optimization is of particular interest for rapidly varying channels, where the energy and bandwidth savings of an optimized scheme can be significant.
In this context, Pilot Symbol Assisted Modulation (PSAM) has emerged as a viable and robust training technique for rapidly varying fading channels. In PSAM, known pilot symbols are multiplexed with data symbols for transmission through the communications channel. At the receiver, knowledge of these pilots is used to form channel estimates, which aid the detection of the data both directly (by modifying the detection rule based on the channel estimate) and indirectly (e.g., by allowing for estimate-directed modulation, power control, and media access). PSAM has been incorporated into standards for IEEE 802.11, Global System for Mobile Communication (GSM), Wideband Code-Division Multiple-Access (WCDMA), and Military protocols, among others, and many theoretical issues are now being addressed. For example, optimized approaches to PSAM have recently been studied from the perspectives of frequency and timing offset estimation, BER, and the channel capacity or its bounds.
Current studies consider PSAM design for continuously time-varying, time-selective Rayleigh fading channels, under the channel capacity or its bounds. In each, the transmitter is assumed to have knowledge of the Doppler spectrum, and the receiver makes (instantaneous) MMSE estimates of the channel based on some subset of the pilot observations. In one study, three estimators (of varying complexity) are proposed and used to predict the channel state for a Gauss-Markov channel correlation model. Optimal binary inputs are used, and it is determined that for sufficiently correlated channels (i.e., slow enough fading), PSAM provides significant gains in the constrained capacity over the no pilot approach. Analysis was carried out through numerical simulation, and the optimization of energy between pilot and data symbols was not attempted.
However, in practice, there exists a need for analytic solutions that provide guidance on the optimal allocation of training and bandwidth. Any such algorithm can be sufficiently simple so as to be amenable to implementation in a consumer grade wireless device, such as a cellphone or personal digital assistant.
In other studies, authors assume a bandlimited Doppler spectrum and derive closed-form bounds on the channel capacity, using the estimator that exploits all past and future pilot observations. Closed-form results are derived for the optimal allocation of training and bandwidth in some cases.
However, in practice, the wireless communications channel available to the system designer is typically digitally sampled, and may not exhibit bandlimited Doppler spectra. Further, practical estimators are limited to using just a few past and future pilots, as using all past and future pilots is prohibitively complex and not feasible in real-world designs. Thus, a heretofore unaddressed need exists in the industry to address the aforementioned inadequacies.